Often teachers and administrators ask if students with special needs should be included in classrooms using Investigations. We agree with the authors of Making Sense (J. Hiebert et. al., Heinemann, 1997) that each student can and has the right to learn mathematics with understanding.
For the past three years we have worked on a project with a dedicated group of classroom and special education teachers from Boston, Brookline, and Cambridge in Massachusetts to identify strategies and principles that will foster the mathematics teaching and learning of students who are struggling.
The Accessible Mathematics Project's collaborative work with teachers resulted in identifying five actions that are critical to teaching meaningful mathematics to students with disabilities in inclusive settings. These actions are:
- Build an inclusive mathematics community based on learning standards-based mathematics
- Make the mathematics explicit
- Expect and support students to work independently and take responsibility
- Link assessment and teaching
- Promote collaboration between special education and classroom teachers
Build an Inclusive Mathematics Community
The first ingredient to put in place in order for students with disabilities to progress in mathematics is to have the teachers expect them to learn along with all of their other students. Students need structures and routines within the mathematics class for working productively together (whole group, working with a partner, independent practice, and closure) as well as procedures for clean-up, organizing and distributing materials, and choosing partners.
Teachers who are successful in inclusive settings find ways to organize their math classes to promote participation from all students. They move among students working in small groups to elicit ideas and check on the understanding of all students. They keep whole group meetings short and active, collecting and displaying students' data or solutions before the meeting in preparation for the discussion, or asking all students to solve problems on individual whiteboards during the meeting and then immediately discussing solutions.
Teachers need to think carefully about grouping their students flexibly according to the demands of the math activity, taking into account students' academic and social skills. Sometimes this means expecting two struggling students to work out their own problem solutions, perhaps directing them to work on one problem that they can present to the class. Sometimes it means pairing a weaker and a stronger student and expecting that the weaker student will be able to explain their solution when they are finished. It is important to provide regular game days and "choice" times or "math menu" activities to meet a variety of students' needs and allow the teachers to work with small groups.
Make the Mathematics Explicit
We found that successful teachers identify ways to make the expectations for tasks and the tasks themselves explicit. They analyze activities for the prerequisite skills and the vocabulary students will need. They work with small groups of students in and outside of class time to pre-teach necessary skills and introduce games and activities ahead of the whole class introduction. During class, they rehearse with students with disabilities ways to show and explain their work so they can take an active role presenting their solutions to the rest of the class.
A crucial element of explicit teaching is helping students know the underlying structure of the mathematics they are doing. This is especially difficult with students who transfer into an Investigations class having little foundational knowledge from which to build any sort of understanding of mathematical relationships and believing that the goal in math class is to reproduce traditional algorithms for computation quickly.
Teachers have observed that some students become confused when they see many different computation strategies. They have found it helpful to model computation strategies that students can make more efficient over time by doing fewer but larger steps (e.g. adding up for subtraction, doing multiplication in parts) and posing problems to elicit these strategies. They provide visual and hands-on models and tools, such as 100 charts, number lines, and connecting cubes for use by students to explain their thinking. They post student solutions for problems such as computation strategies and lists of factors so that the students are able to make connections with prior work.
Expect and Support Students to Work Independently and Take Responsibility
A third ingredient is that teachers need to help students "learn how to learn." Often students with disabilities are dependent on adults for help and don't expect to be able to make sense of the math. When a student asks for assistance, successful teachers help the student articulate what he or she already knows and what is confusing. They encourage students to evaluate and check their own work, either on their own or by working with another student.
Teachers whose students thrive in inclusive settings also find it essential to help students develop structures for organizing their work and recording their solutions. They may "take dictation" by listening to a child explain their solution and modeling how to write down the child's solution in an organized way.
Link Assessment and Teaching
A fourth ingredient for success with students with disabilities is the necessity of linking ongoing assessment and teaching. Teachers need to pose questions regularly that engage the child and provide information about the student's understanding. They annotate student work to keep track of student progress. Sometimes they use assessment instruments that reveal how children are developing mathematical ideas, looking especially for what each child knows. For example, the teachers in the Accessible Math Project administered parts of the Early Numeracy Assessment 1 about number, time, and space (geometry) to some of their students and planned instruction based on the strengths and gaps in the students' knowledge.
Teachers acknowledge and accept differences in how students learn, what their strengths are and what they find difficult. Using mistakes and confusions as focal points for learning helps students to work together and to feel safe taking risks.
Some teachers ask students to evaluate a set of work from the class or of similar work from another class. They pose questions to help students develop criteria for evaluating and comparing the work. Is the response organized so that you can follow the student's steps? What could the student do to help you better understand how she arrived at the answer? If you had to choose one of these procedures to do subtraction problems, which would you choose? How could you adapt this procedure for problems with larger numbers?
This fifth ingredient emerged from analysis of the data of Karen Mutch-Jones based on her work with the project's teachers.2 Data indicate that collaboration, which has been a strong focus of the teachers involved in the project, between special education and classroom teachers contributes to the formation of a mathematical community. Her data reveals that impacts of collaboration on the community include:
- All students form a relationship with and seek help from both teachers instead of seeing the special education teachers as the teacher for those kids.
- Expectations for learning behavior (e.g. paying attention, participating in the group, quietly moving away from the group when necessary) during math class are the same for all students.
- Teachers help each other to establish fair, high expectations for learning mathematics for all students.
- All students have access to a standards-based curriculum, to learn mathematics concepts with understanding and to develop strong problem-solving strategies.
Her findings also show that collaboration provided the following opportunities for teachers:
- Allowed a broader or deeper understanding of mathematics content and/or the Investigations curriculum.
- Learned to ask each other and their students better questions about mathematical thinking and math curriculum because of collaboration.
- Provided expanded ways of thinking about student abilities and needs.
Finally, an additional form of collaboration that our project has found to be important is the collaborative relationship formed between the teachers and the classroom support staff (para-educators and student teachers). Too often the support staff are not trained in the mathematics curriculum so their role during math class is limited.
Cornelia Tierney and Judy Storeygard, TERC
1. A one-on-one interview developed by the Early Numeracy Research Project (ENRP) was established in 1999 as a joint venture between Australian Catholic University, Monash University, the Victorian Department of Employment, Education and Training (DEET), the Catholic Education Office (CEO, Melbourne), and the Association of Independent Schools Victoria (AISV).
2. Karen Mutch-Jones is a colleague whose doctoral dissertation (June, 2004) from the Harvard University Graduate School of Education, "Collaborative Insights: The Work of General and Special Educator Pairs in Inclusive Mathematics Classrooms" is based on her work on this project.
This content of this Spotlight article is based on work from The Accessible Mathematics Project which was supported by the National Science Foundation (NSF grant no. HRD--0090070)